On a spectral sequence for twisted cohomologies

Abstract

Let (Ω*(M), d) be the de Rham cochain complex for a smooth compact closed manifolds M of dimension n. For an odd-degree closed form H, there is a twisted de Rham cochain complex (Ω*(M), d + H _∧) and its associated twisted de Rham cohomology H*(M,H). The authors show that there exists a spectral sequence {E _r ^p,q, d _r} derived from the filtration F_p (\Omega ^ * (M)) = \mathop \oplus \limits_{i \geqslant p} \Omega ^i (M) of Ω*M, which converges to the twisted de Rham cohomology H*(M,H). It is also shown that the differentials in the spectral sequence can be given in terms of cup products and specific elements of Massey products as well, which generalizes a result of Atiyah and Segal. Some results about the indeterminacy of differentials are also given in this paper.

Keywords

Spectral sequence / Twisted de Rham cohomology / Massey product / Differential

References

[1] Atiyah M, Segal G B. Twisted K-Theory and Cohomology, Inspired by S. S. Chern, 2006, Hackensack, NJ: World Sci. Publ. 5-43

[2] Deligne P, Griffiths P, Morgan J, Sullivan D. Real homotopy theory of Kähler manifolds. Invent. Math., 1975, 29(3): 245-274

[3] Hatcher A. Algebraic Topology, 2002, Cambridge: Cambridge University Press

[4] Kraines D. Massey higher products. Trans. Amer. Math. Soc., 1966, 124: 431-449

[5] Kraines D, Schochet C. Differentials in the Eilenberg-Moore spectral sequence. J. Pure Appl. Algebra, 1972, 2(2): 131-148

[6] Massey W S. Exact couples in algebraic topology, I, II. Ann. of Math. (2), 1952, 56: 363-396

[7] Massey W S. Exact couples in algebraic topology, III, IV, V. Ann. of Math. (2), 1953, 57: 248-286

[8] Massey W S. Some higher order cohomology operations. 1958 Symposium Internacional de Topologá Algebraica International Symposium on Algebraic Topology, 1958, Mexico: Universidad Nacional Autónoma de México and Unesco 145-154

[9] Mathai V, Wu S. Analytic torsion for twisted de Rham complexes. J. Diff. Geom., 2011, 88(2): 297-332

[10] May J P. Matric Massey products. J. Algebra, 1969, 12: 533-568

[11] May J P. The cohomology of augmented algebras and generalized Massey products for DGA-algebras. Trans. Amer. Math. Soc., 1966, 122: 334-340

[12] McCleary J. A User’s Guide to Spectral Sequences, 2001 2nd Edition Cambridge: Cambridge University Press

[13] Rohm R, Witten E. The antisymmetric tensor field in superstring theory. Ann. Physics, 1986, 170(2): 454-489

[14] Sharifi R T. Massey products and ideal class groups. J. Reine Angew. Math., 2007, 603: 1-33

[15] Spanier E W. Algebraic Topology, 1981, New York, Berlin: Springer-Verlag